A282035 -id:A282035 - cp's OEIS frontend

A076409 Sum of the quadratic residues of prime(n).

1, 1, 5, 7, 22, 39, 68, 76, 92, 203, 186, 333, 410, 430, 423, 689, 767, 915, 1072, 994, 1314, 1343, 1577, 1958, 2328, 2525, 2369, 2675, 2943, 3164, 3683, 3930, 4658, 4587, 5513, 5134, 6123, 6520, 6012, 7439, 7518, 8145, 7831, 9264, 9653, 8955, 10761, 11596

Offset: 1

R. K. Guy, Oct 08 2002

Row sums of A063987. - R. J. Mathar, Jan 08 2015

prime(n) divides a(n) for n > 2. This is implied by a variant of Wolstenholme's theorem (see Hardy & Wright reference). - Isaac Saffold, Jun 21 2018

			If n = 3, then p = 5 and a(3) = 1 + 4 = 5. If n = 4, then p = 7 and a(4) = 1 + 4 + 2 = 7. If n = 5, then p = 11 and a(5) = 1 + 4 + 9 + 5 + 3 = 22. - _Michael Somos_, Jul 01 2018

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 4th ed., Oxford Univ. Press, 1960, p. 88-90.
Kenneth A. Ribet, Modular forms and Diophantine questions, Challenges for the 21st century (Singapore 2000), 162-182; World Sci. Publishing, River Edge NJ 2001; Math. Rev. 2002i:11030.

Zak Seidov, Table of n, a(n) for n = 1..1000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).

Cf. A076410.

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

A076409 := proc(n)
  local a,p,i ;
  p := ithprime(n) ;
  a := 0 ;
  for i from 1 to p-1 do
    if numtheory[legendre](i,p) = 1 then
       a := a+i ;
    end if;
  end do;
  a ;
end proc: # R. J. Mathar, Feb 26 2011

Join[{1,1}, Table[ Apply[ Plus, Flatten[ Position[ Table[ JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], {n, 3, 48}]]
Join[{1}, Table[p=Prime[n]; If[Mod[p,4]==1, p(p-1)/4, Sum[PowerMod[k,2, p],{k,p/2}]], {n,2,1000}]] (* Zak Seidov, Nov 02 2011 *)
a[ n_] := If[ n < 3, Boole[n > 0], With[{p = Prime[n]}, Sum[ Mod[k^2, p], {k, (p - 1)/2}]]]; (* Michael Somos, Jul 01 2018 *)

a(n,p=prime(n))=if(p<5,return(1)); if(k%4==1, return(p\4*p)); sum(k=1,p-1,k^2%p) \\ Charles R Greathouse IV, Feb 21 2017

If prime(n) = 4k+1 then a(n) = k*(4k+1).

For n>2 if prime(n) = 4k+3 then a(n) = (k - b)*(4k+3) where b = (h(-p) - 1) / 2; h(-p) = A002143. For instance. If n=5, p=11, k=2, b=(1-1)/2=0 and a(5) = 2*11 = 22. If n=20, p=71, k=17, b=(7-1)/2=3 and a(20) = 14*71 = 994. - Andrés Ventas, Mar 01 2021

Edited and extended by Robert G. Wilson v, Oct 09 2002

A282043 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic nonresidues mod p.

Original entry on oeis.org

14, 161, 279, 658, 1491, 1738, 2884, 4318, 6191, 7849, 10314, 10746, 13157, 16013, 18936, 19783, 27057, 35541, 35232, 39832, 50858, 51363, 55097, 63228, 60875, 68408, 97038, 95906, 103484, 111931, 140205, 136676, 145628, 146445, 172830, 189614, 195038, 209332, 221373, 219641, 238849, 254597

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282042 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 7 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282039, A282040, A282041, A282039 again, A282042, A282043

sqnr[p_] := Select[Range[p-1], JacobiSymbol[#, p] != 1&] // Total;
sqnr /@ Select[Prime[Range[200]], Mod[#, 8] == 7&] (* Jean-François Alcover, Aug 30 2018 *)

A282721 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.

Original entry on oeis.org

1, 13, 32, 137, 306, 314, 555, 876, 1400, 1416, 1742, 2450, 3099, 3788, 4816, 5430, 6351, 7344, 8393, 9546, 12858, 13373, 15265, 17277, 16311, 18403, 19521, 22344, 21805, 23590, 25495, 26805, 30767, 30863, 31570, 35980, 40678, 43946, 45640, 49124, 50055, 52776, 58418, 66210, 71521, 71665, 83666, 81628

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..4000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Cf. A282035-A282043 and A282722-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
  q:=q+j;
  if jA282721 - A282727
# Alternative
f:= proc(p) local q,r,t,j;
  r:= (p-1)/2; t:= 0;
  for j from 1 to r do
    q:= j^2 mod p;
    if q <= r then t:= t+q fi;
od:
t
end proc:
map(f, select(isprime, [seq(i,i=3..10000,8)])); # Robert Israel, Mar 27 2017

s[p_] := Total[Select[Range[Floor[p/2]], JacobiSymbol[#, p] == 1&]];
s /@ Select[Range[3, 2000, 8], PrimeQ] (* Jean-François Alcover, Nov 17 2017 *)

from sympy import isprime
def a(p):
    r=(p - 1)//2
    t=0
    for j in range(1, r + 1):
        q=(j**2)%p
        if q<=r:t+=q
    return t
print([a(p) for p in range(3, 2001, 8) if isprime(p)]) # Indranil Ghosh, Mar 27 2017, translated from Maple code

A282727 Let p = n-th prime == 3 mod 8; a(n) = (sum of quadratic residues mod p that are < p/2) + (sum of all quadratic residues mod p).

Original entry on oeis.org

2, 35, 108, 567, 1073, 1386, 2132, 3551, 5330, 6003, 8262, 9968, 13860, 16046, 19625, 24957, 29376, 34155, 37541, 44793, 54758, 61217, 68036, 75215, 77688, 85347, 93366, 98912, 101745, 107531, 119583, 129042, 135548, 145607, 149040, 170478, 193356, 205335, 213521, 230373, 243432, 256851, 280016, 294395

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

This is also the (sum of quadratic nonresidues mod p that are < p/2) + (sum of all quadratic nonresidues mod p) (= A282721 + A282723 = A282724 + A282726).

Robert Israel, Table of n, a(n) for n = 1..4000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Cf. A282035-A282043 and A282721-A282726.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[]; Th:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 3 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
q:=q+j;
if jA282721 - A282727
# Alternative:
v:= proc(x,r) if x <= r then 2*x else x fi end proc:
f:= proc(p) local q, r, j;
  r:= (p-1)/2;
  add(v(j^2 mod p, r), j=1..r)
end proc:
map(f, select(isprime, [seq(i,i=3..1000,8)])); # Robert Israel, Mar 27 2017

v[x_, r_] := If[x <= r, 2*x, x];
f[p_] := Module[{r}, r = (p-1)/2; Sum[v[PowerMod[j, 2, p], r], {j, 1, r}]];
f /@ Select[Range[3, 1000, 8], PrimeQ] (* Jean-François Alcover, Nov 27 2017, after Robert Israel *)

from sympy import isprime
def v(x, r):
    return 2*x if x<=r else x
def a(p):
    r=(p - 1)//2
    return sum(v((j**2)%p, r) for j in range(1, r + 1))
print([a(p) for p in range(3, 2001, 8) if isprime(p)]) # Indranil Ghosh, Mar 27 2017 translated from  Robert Israel's Maple program

A125615 Sum of the quadratic nonresidues of prime(n).

Original entry on oeis.org

0, 2, 5, 14, 33, 39, 68, 95, 161, 203, 279, 333, 410, 473, 658, 689, 944, 915, 1139, 1491, 1314, 1738, 1826, 1958, 2328, 2525, 2884, 2996, 2943, 3164, 4318, 4585, 4658, 5004, 5513, 6191, 6123, 6683, 7849, 7439, 8413, 8145, 10314, 9264, 9653, 10746, 11394

Offset: 1

Nick Hobson, Nov 30 2006

For all n > 2, prime(n) divides a(n).

			The quadratic nonresidues of 7=prime(4) are 3, 5 and 6. Hence a(4) = 3+5+6 = 14.

D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 185.

Nick Hobson, Table of n, a(n) for n = 1..1000
Christian Aebi and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT], 2015.

Cf. A076409, A076410, A125613-A125618.

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Table[p=Prime[n];Total[Complement[Range[p-1],Union[Table[PowerMod[k, 2, p], {k, p}]]]],{n,47}] (* James C. McMahon, Dec 19 2024 *)

vector(47, n, p=prime(n); t=1; for(i=2, (p-1)/2, t+=((i^2)%p)); p*(p-1)/2-t)

If prime(n) = 4k+1 then a(n) = k(4k+1) = A076409(n).

A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

Original entry on oeis.org

5, 39, 68, 203, 333, 410, 689, 915, 1314, 1958, 2328, 2525, 2943, 3164, 4658, 5513, 6123, 7439, 8145, 9264, 9653, 13053, 13514, 14460, 16448, 18023, 19113, 19670, 21389, 24414, 25043, 28308, 30363, 31064, 34689, 37733, 39303, 40100, 41718, 44205, 46764, 50288

Offset: 1

Juri-Stepan Gerasimov, Dec 11 2009

nonn

The halves of even numbers of the form p(p-1)/2 for p prime.

Sum of the quadratic residues of primes of the form 4k + 1. For example, a(3)=68 because 17 is the 3rd prime of the form 4k + 1 and the quadratic residues of 17 are 1, 4, 9, 16, 8, 2, 15, 13 which sum to 68. This sum is also the sum of the quadratic nonresidues. Cf. A230077. - Geoffrey Critzer, May 07 2015

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.21 p. 110.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A005098, A007742, A008837.

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Table[Table[Mod[a^2, p], {a, 1, (p - 1)/2}] // Total, {p,
Select[Prime[Range[100]], Mod[#, 4] == 1 &]}] (* Geoffrey Critzer, May 07 2015 *)
Select[(# (#-1))/4&/@Prime[Range[100]],IntegerQ] (* Harvey P. Dale, Dec 24 2022 *)

lista(nn) = forprime(p=2, nn, if ((p % 4)==1, print1(p*(p-1)/4, ", "))); \\ Michel Marcus, Mar 23 2016

Corrected (16448 inserted, 25043 inserted) by R. J. Mathar, May 22 2010

A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

Original entry on oeis.org

2, 14, 33, 95, 161, 279, 473, 658, 944, 1139, 1491, 1738, 1826, 2884, 2996, 4318, 4585, 5004, 6191, 6683, 7849, 8413, 10314, 10746, 11394, 13157, 13393, 16013, 16566, 18936, 19783, 20376, 23946, 27057, 27804, 30883, 35541, 35232, 36384, 39832, 45671, 50858, 51363, 50059, 55097, 56040

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Christian Aebi, Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv:1512.00896 [math.NT], 2015.

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

Cf. A002145.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037
# Alternative:
f:= p -> add(-k^2 mod p, k=1..(p-1)/2)::
map(f, select(isprime, [seq(p,p=3..1000,4)])); # Robert Israel, Nov 09 2020

f[p_] := Total[Range[p-1] ~Complement~ Table[Mod[k^2, p], {k, (p-1)/2}] ]; f /@ Select[Range[3, 1000, 4], PrimeQ] (* Jean-François Alcover, Feb 16 2018, after Robert Israel *)

lista(nn) = forprime(p=2, nn, if(p%4==3, print1(sum(k=1, p-1, if (!issquare(Mod(k, p)), k)), ", "))); \\ Michel Marcus, Nov 09 2020

a(n) = Sum_{k=1..(A002145(n)-1)/2} (-k^2) mod A002145(n). - J. M. Bergot and Robert Israel, Nov 09 2020

A282037 Let p = n-th prime == 3 mod 4; a(n) = (sum of quadratic nonresidues mod p) - (sum of quadratic residues mod p).

Original entry on oeis.org

1, 7, 11, 19, 69, 93, 43, 235, 177, 67, 497, 395, 249, 515, 321, 635, 655, 417, 1057, 163, 1837, 895, 2483, 1791, 633, 1561, 1135, 3585, 1757, 3419, 2981, 849, 921, 5909, 993, 1735, 6821, 3303, 1137, 6511, 3771, 9051, 6585, 2215, 3241, 3269, 11975, 3409, 4419, 1497, 10563, 2615, 1641, 5067, 2855

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Equals A282036 - A282035.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Christian Aebi and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).

Sums of residues, nonresidues, and their differences, for p == 1 mod 4, p == 3 mod 4, and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 200 do
p:=ithprime(i1);
if (p mod 4) = 3 then
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
fi;
od:
a; m; d; # A282035, A282036, A282037

sum[p_] := Total[If[JacobiSymbol[#, p] == 1, -#, #]& /@ Range[p-1]];
sum /@ Select[Prime[Range[200]], Mod[#, 4] == 3&] (* Jean-François Alcover, Aug 31 2018 *)

A282038 (Sum of the quadratic nonresidues of prime(n)) - (sum of the quadratic residues of prime(n)).

Original entry on oeis.org

-1, 1, 0, 7, 11, 0, 0, 19, 69, 0, 93, 0, 0, 43, 235, 0, 177, 0, 67, 497, 0, 395, 249, 0, 0, 0, 515, 321, 0, 0, 635, 655, 0, 417, 0, 1057, 0, 163, 1837, 0, 895, 0, 2483, 0, 0, 1791, 633, 1561, 1135, 0, 0, 3585, 0, 1757, 0, 3419, 0, 2981, 0, 0, 849, 0, 921, 5909, 0, 0, 993, 0, 1735, 0, 0, 6821, 3303, 0

Offset: 1

N. J. A. Sloane, Feb 20 2017

sign

Equals 0 if p == 1 (mod 4).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Christian Aebi and Grant Cairns, Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 [math.NT] (2015).

Sums of residues, nonresidues, and their differences, for p == 1 (mod 4), p == 3 (mod 4), and all p: A171555; A282035, A282036, A282037; A076409, A125615, A282038.

with(numtheory):
a:=[]; m:=[]; d:=[];
for i1 from 1 to 100 do
p:=ithprime(i1);
sp:=0; sm:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then sp:=sp+j; else sm:=sm+j; fi; od;
a:=[op(a),sp]; m:=[op(m),sm]; d:=[op(d),sm-sp];
od:
a; m; d; # A076409, A125615, A282038

sum[p_] := Total[If[JacobiSymbol[#, p] == 1, -#, #]& /@ Range[p-1]];
a[n_] := sum[Prime[n]];
Array[a, 100] (* Jean-François Alcover, Aug 31 2018 *)

a(n) = my (p=prime(n)); return (sum(i=1, p-1, if (kronecker(i,p)==1, -i, +i))) \\ Rémy Sigrist, Apr 28 2017

A282039 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.

Original entry on oeis.org

3, 33, 60, 138, 315, 390, 663, 1008, 1425, 1743, 2280, 2475, 3108, 3570, 4323, 4590, 6045, 8055, 8418, 9168, 11610, 12045, 13398, 14340, 14823, 15813, 22425, 23028, 24885, 26163, 32310, 33033, 34503, 35250, 42333, 43995, 46548, 49173, 51870, 52785, 58443, 60393, 61380, 66435, 67470, 70623

Offset: 1

N. J. A. Sloane, Feb 20 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Aebi, Christian, and Grant Cairns. Sums of Quadratic residues and nonresidues, arXiv preprint arXiv:1512.00896 (2015).

Cf. A282035-A282043 and A282721-A282727.

with(numtheory):
Ql:=[]; Qu:=[]; Q:=[]; Nl:=[]; Nu:=[]; N:=[];
for i1 from 1 to 300 do
p:=ithprime(i1);
if (p mod 8) = 7 then
ql:=0; qu:=0; q:=0; nl:=0; nu:=0; n:=0;
for j from 1 to p-1 do
if legendre(j,p)=1 then
  q:=q+j;
  if jA282039, A282040, A282041, A282039 again, A282042, A282043
# alternative:
g:= proc(t,p) if t < p/2 then t else 0 fi end proc;
f:= proc(n) local k;
     add(g(k^2 mod n, n),k=1..n/2)
end proc:
P:= select(isprime, [seq(i,i=7..3000,8)]):
map(f,P); # Robert Israel, Nov 09 2020

sum[p_]:= Total[If[#Jean-François Alcover, Aug 31 2018 *)

cp's OEIS Frontend

A076409 Sum of the quadratic residues of prime(n).

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A282043 Let p = n-th prime == 7 mod 8; a(n) = sum of quadratic nonresidues mod p.

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A282721 Let p = n-th prime == 3 mod 8; a(n) = sum of quadratic residues mod p that are < p/2.

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A282727 Let p = n-th prime == 3 mod 8; a(n) = (sum of quadratic residues mod p that are < p/2) + (sum of all quadratic residues mod p).

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A125615 Sum of the quadratic nonresidues of prime(n).

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A171555 Numbers of the form prime(n)*(prime(n)-1)/4.

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A282036 a(n) is the sum of quadratic nonresidues of A002145(n) (the n-th prime == 3 mod 4).

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